Group #20253968 — subgroups

Subgroups — #20253968

Order 6016.

456 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#15689674	C₂	1	2	yes	maximal, complemented
2	#15689500	C₂	1	1	yes	maximal, characteristic, complemented
3	#15689471	C₂	1	2	yes	maximal, complemented
4	#15689360	C₂	1	4	yes	maximal, complemented
5	#15689205	C₂	1	4	yes	maximal, complemented
6	C₄×C₈×D₄₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#500	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10625354	—	2	2	no	—
9	C₂×D₃₇₆	C₂²	1	4	yes	—
10	C₂×C₈×D₄₇	C₂²	1	2	yes	—
11	#13094021	C₂²	1	2	yes	—
12	#13094013	—	2	2	no	—
13	#13093999	—	2	2	no	—
14	#13093999	C₂²	1	4	yes	—
15	#13093986	—	2	2	no	—
16	#13093986	C₂²	1	4	yes	—
17	C₄²×D₄₇	C₂²	1	1	yes	characteristic
18	#13093944	C₂²	1	8	yes	—
19	C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
20	C₈×Dic₄₇	C₂²	1	2	yes	—
21	C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
22	#107×C₄₇	C₂²	1	4	yes	—
23	C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
24	C₂×Q₈×C₄₇	—	2	2	no	—
25	#10625354	—	4	8	no	—
26	C₂×D₁₈₈	—	2	4	no	—
27	C₂×D₁₈₈	—	2	2	no	—
28	C₂×D₁₈₈	—	2	2	no	—
29	C₂×D₁₈₈	C₂³	1	4	yes	—
30	C₂×C₄×D₄₇	—	2	4	no	—
31	C₂×C₄×D₄₇	D₄	1	1	yes	characteristic
32	C₂×C₄×D₄₇	D₄	1	1	yes	characteristic
33	C₂×C₄×D₄₇	C₂³	1	1	yes	characteristic
34	C₂×C₈×C₄₇	C₂³	1	2	yes	—
35	C₄⋊C₄×C₄₇	—	2	2	no	—
36	C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
37	C₄²×C₄₇	C₂³	1	1	yes	characteristic
38	(C₂×D₄₇)⋊C₄	—	2	4	no	—
39	(C₂×D₄₇)⋊C₄	—	2	4	no	—
40	C₄×Dic₄₇	D₄	1	1	yes	characteristic
41	C₄×Dic₄₇	D₄	1	1	yes	characteristic
42	C₄×Dic₄₇	C₂³	1	1	yes	characteristic
43	C₂×C₄₇⋊C₈	C₂³	1	2	yes	—
44	D₃₇₆	—	2	8	no	—
45	C₈×D₄₇	—	2	4	no	—
46	C₂²×D₄₇	—	2	4	no	—
47	C₂²×D₄₇	C₂×D₄	1	1	yes	characteristic
48	Q₈×C₄₇	—	4	4	no	—
49	C₂×C₄×C₄₇	—	2	4	no	—
50	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
53	C₂×Dic₄₇	C₂×D₄	1	1	yes	characteristic
54	C₂×Dic₄₇	C₂×D₄	1	1	yes	characteristic
55	C₂×Dic₄₇	C₂×D₄	1	1	yes	characteristic
56	D₁₈₈	—	2	8	no	—
57	D₁₈₈	—	4	8	no	—
58	D₁₈₈	—	4	4	no	—
59	D₁₈₈	—	4	4	no	—
60	C₄×D₄₇	—	2	2	no	—
61	C₄×D₄₇	—	2	2	no	—
62	C₄×D₄₇	—	4	8	no	—
63	C₄×D₄₇	D₈	1	4	yes	complemented
64	C₈×C₄₇	#41	1	4	yes	complemented
65	C₄₇⋊C₈	—	2	2	no	—
66	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
67	C₂×D₄₇	—	2	1	no	—
68	C₂×D₄₇	—	2	1	no	—
69	C₂×D₄₇	—	4	4	no	—
70	C₂×D₄₇	—	4	4	no	—
71	C₂×D₄₇	—	4	4	no	—
72	C₂×D₄₇	C₂×D₈	1	2	yes	—
73	C₄×C₄₇	—	4	4	no	—
74	C₄×C₄₇	C₂×D₈	1	2	yes	—
75	C₄×C₄₇	#135	1	2	yes	—
76	C₄×C₄₇	C₂×#41	1	2	yes	—
77	Dic₄₇	—	2	1	no	—
78	Dic₄₇	—	2	1	no	—
79	Dic₄₇	C₂×D₈	1	2	yes	—
80	#3169	—	47	1	no	maximal, Sylow
81	C₂×C₄₇	#532	1	1	yes	characteristic
82	C₂×C₄₇	C₂²×D₈	1	1	yes	characteristic
83	C₂×C₄₇	C₂×#135	1	1	yes	characteristic
84	D₄₇	—	2	2	no	—
85	D₄₇	—	8	4	no	—
86	#532	—	47	2	no	—
87	#500	D₄₇	1	1	yes	characteristic, complemented
88	#493	—	47	1	no	—
89	#486	—	47	2	no	—
90	#466	—	47	4	no	—
91	C₄×D₈	—	47	4	no	—
92	C₂×C₄×C₈	—	47	1	no	—
93	C₄₇	#3169	1	1	yes	characteristic, complemented, Sylow
94	C₂×#41	—	94	2	no	—
95	C₂×D₈	—	47	4	no	—
96	C₂²×C₈	—	47	2	no	—
97	C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
98	#127	—	47	2	no	—
99	#124	—	94	2	no	—
100	#121	—	47	4	no	—
101	#121	—	94	2	no	—
102	C₄×D₄	—	47	4	no	—
103	C₄×D₄	—	94	2	no	—
104	C₂×C₄²	—	47	1	no	—
105	#107	C₂×D₄₇	1	4	yes	—
106	#102	—	47	8	no	—
107	C₄×C₈	—	47	2	no	—
108	C₄×C₈	—	47	1	no	—
109	C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
110	#41	—	188	8	no	—
111	C₂×Q₈	—	2	2	no	—
112	C₂×D₄	—	47	4	no	—
113	C₂×D₄	—	94	4	no	—
114	C₂×D₄	—	94	2	no	—
115	C₂×D₄	—	94	2	no	—
116	C₂²×C₄	—	47	1	no	—
117	C₂²×C₄	—	47	1	no	—
118	C₂²×C₄	—	47	1	no	—
119	C₂²×C₄	—	94	4	no	—
120	D₈	—	94	8	no	—
121	C₂×C₈	—	47	2	no	—
122	C₂×C₈	—	94	4	no	—
123	C₂×C₈	C₂²×D₄₇	1	2	yes	—
124	C₄⋊C₄	—	2	2	no	—
125	C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
126	C₂²⋊C₄	—	94	4	no	—
127	C₂²⋊C₄	—	94	4	no	—
128	C₄²	—	47	1	no	—
129	C₄²	—	47	1	no	—
130	C₄²	—	47	1	no	—
131	C₄²	C₂²×D₄₇	1	1	yes	characteristic
132	C₂³	—	47	1	no	—
133	C₂³	—	94	4	no	—
134	Q₈	—	4	4	no	—
135	D₄	—	94	8	no	—
136	D₄	—	188	8	no	—
137	D₄	—	188	4	no	—
138	D₄	—	188	4	no	—
139	C₂×C₄	—	2	4	no	—
140	C₂×C₄	—	47	1	no	—
141	C₂×C₄	—	47	4	no	—
142	C₂×C₄	—	47	1	no	—
143	C₂×C₄	—	47	1	no	—
144	C₂×C₄	—	94	2	no	—
145	C₂×C₄	—	94	2	no	—
146	C₂×C₄	—	188	8	no	—
147	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
148	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
149	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
150	C₈	—	94	2	no	—
151	C₈	#10625354	1	4	yes	complemented
152	C₂²	—	47	2	no	—
153	C₂²	—	94	1	no	—
154	C₂²	—	94	1	no	—
155	C₂²	—	188	4	no	—
156	C₂²	—	188	4	no	—
157	C₂²	—	188	4	no	—
158	C₂²	C₂×D₄×D₄₇	1	1	yes	characteristic
159	C₄	—	4	4	no	—
160	C₄	—	47	2	no	—
161	C₄	—	94	1	no	—
162	C₄	—	94	1	no	—
163	C₄	D₈×D₄₇	1	2	yes	—
164	C₄	#13094047	1	2	yes	—
165	C₄	C₂×#10625354	1	2	yes	—
166	C₂	—	94	2	no	—
167	C₂	—	376	4	no	—
168	C₂	#15689674	1	1	yes	characteristic
169	C₂	C₂×D₈×D₄₇	1	1	yes	characteristic
170	C₂	C₂×#13094047	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689674

C₂

yes

maximal, complemented

#15689500

C₂

yes

maximal, characteristic, complemented

#15689471

C₂

yes

maximal, complemented

#15689360

C₂

yes

maximal, complemented

#15689205

C₂

yes

maximal, complemented

C₄×C₈×D₄₇

C₂

yes

maximal, characteristic, complemented

C₄₇×#500

C₂

yes

maximal, characteristic, complemented

C₂×#10625354

—

C₂×D₃₇₆

C₂²

yes

—

C₂×C₈×D₄₇

C₂²

yes

—

#13094021

C₂²

yes

—

#13094013

—

#13093999

—

#13093999

C₂²

yes

—

#13093986

—

#13093986

C₂²

yes

—

C₄²×D₄₇

C₂²

yes

characteristic

#13093944

C₂²

yes

—

C₄×C₄₇⋊C₈

C₂²

yes

characteristic

C₈×Dic₄₇

C₂²

yes

—

C₄⋊Q₈×C₄₇

C₂²

yes

—

#107×C₄₇

C₂²

yes

—

C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂×Q₈×C₄₇

—

#10625354

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

C₂³

yes

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

D₄

yes

characteristic

C₂×C₄×D₄₇

D₄

yes

characteristic

C₂×C₄×D₄₇

C₂³

yes

characteristic

C₂×C₈×C₄₇

C₂³

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

C₂³

yes

—

C₄²×C₄₇

C₂³

yes

characteristic

(C₂×D₄₇)⋊C₄

—

(C₂×D₄₇)⋊C₄

—

C₄×Dic₄₇

D₄

yes

characteristic

C₄×Dic₄₇

D₄

yes

characteristic

C₄×Dic₄₇

C₂³

yes

characteristic

C₂×C₄₇⋊C₈

C₂³

yes

—

D₃₇₆

—

C₈×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

C₂×D₄

yes

characteristic

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₂×Dic₄₇

C₂×D₄

yes

characteristic

C₂×Dic₄₇

C₂×D₄

yes

characteristic

C₂×Dic₄₇

C₂×D₄

yes

characteristic

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

C₄×D₄₇

—

C₄×D₄₇

—

C₄×D₄₇

—

C₄×D₄₇

D₈

yes

complemented

C₈×C₄₇

#41

yes

complemented

C₄₇⋊C₈

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

C₂×D₈

yes

—

C₄×C₄₇

—

C₄×C₄₇

C₂×D₈

yes

—

C₄×C₄₇

#135

yes

—

C₄×C₄₇

C₂×#41

yes

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

C₂×D₈

yes

—

—

C₂×C₄₇

yes

C₂×C₄₇

C₂²×D₈

yes

characteristic

C₂×C₄₇

C₂×#135

yes

characteristic

D₄₇

—

D₄₇

—

#532

—

#500

D₄₇

yes

characteristic, complemented

#493

—

#486

—

#466

—

C₄×D₈

—

C₂×C₄×C₈

—

C₄₇

#3169

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂×D₈

—

C₂²×C₈

—

C₄⋊Q₈

C₂×D₄₇

yes

—

#127

—

#124

—

100

#121

—

101

#121

—

102

C₄×D₄

—

103

C₄×D₄

—

104

C₂×C₄²

—

105

#107

C₂×D₄₇

yes

—

106

#102

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₄×C₈

C₂×D₄₇

yes

characteristic

110

#41

—

188

—

111

C₂×Q₈

—

112

C₂×D₄

—

113

C₂×D₄

—

114

C₂×D₄

—

115

C₂×D₄

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

D₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

C₂²×D₄₇

yes

—

124

C₄⋊C₄

—

125

C₄⋊C₄

C₂²×D₄₇

yes

—

126

C₂²⋊C₄

—

127

C₂²⋊C₄

—

128

C₄²

—

129

C₄²

—

130

C₄²

—

131

C₄²

C₂²×D₄₇

yes

characteristic

132

C₂³

—

133

C₂³

—

134

Q₈

—

135

D₄

—

136

D₄

—

188

—

137

D₄

—

188

—

138

D₄

—

188

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

188

—

147

C₂×C₄

D₄×D₄₇

yes

characteristic

148

C₂×C₄

D₄×D₄₇

yes

characteristic

149

C₂×C₄

C₂³×D₄₇

yes

characteristic

150

C₈

—

151

C₈

#10625354

yes

complemented

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

188

—

156

C₂²

—

188

—

157

C₂²

—

188

—

158

C₂²

C₂×D₄×D₄₇

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

D₈×D₄₇

yes

—

164

C₄

#13094047

yes

—

165

C₄

C₂×#10625354

yes

—

166

C₂

—

167

C₂

—

376

—

168

C₂

#15689674

yes

characteristic

169

C₂

C₂×D₈×D₄₇

yes

characteristic

170

C₂

C₂×#13094047

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₇
Fitting	C₄₇×#500	C₂
socle	C₂²×C₄₇	C₂²×D₄
hypercenter	#500	D₄₇
hypocenter	C₄₇	#3169
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

subgroup

G/H

center

C₂²

C₂×D₄×D₄₇

commutator

C₂×C₄×C₄₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₇

Fitting

C₄₇×#500

C₂

socle

C₂²×C₄₇

C₂²×D₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #500

N	H ≅ G/N	type
1. #15689674	C₂	semidirect
2. #15689500	C₂	semidirect
3. #15689471	C₂	semidirect
4. #15689360	C₂	semidirect
5. #15689205	C₂	semidirect
6. C₄×C₈×D₄₇	C₂	semidirect
7. C₄₇×#500	C₂	semidirect
63. C₄×D₄₇	D₈	semidirect
64. C₈×C₄₇	#41	semidirect
87. #500	D₄₇	semidirect
93. C₄₇	#3169	semidirect
151. C₈	#10625354	semidirect

H ≅ G/N

type

1. #15689674

C₂

semidirect

2. #15689500

C₂

semidirect

3. #15689471

C₂

semidirect

4. #15689360

C₂

semidirect

5. #15689205

C₂

semidirect

6. C₄×C₈×D₄₇

C₂

semidirect

7. C₄₇×#500

C₂

semidirect

63. C₄×D₄₇

D₈

semidirect

64. C₈×C₄₇

#41

semidirect

87. #500

D₄₇

semidirect

93. C₄₇

#3169

semidirect

151. C₈

#10625354

semidirect

Subgroups — #20253968

Conjugacy classes of subgroups

Complements